Question

A metallic rod of length $l$ is tied to a string of length $2l$ and made to rotate with angular speed $\omega$ on a horizontal table with one end of the string fixed at point $\text{O}$. If there is a vertical magnetic field $B$ in the region, the e.m.f induced across the ends of the rod is:

• A. $\frac{3B\omega l^2}{2}$
• B. $\frac{4B\omega l^2}{2}$
• C. $\frac{5B\omega l^2}{2}$
• D. $\frac{2B\omega l^2}{2}$

Answer 1

The correct option is C $\frac{5B\omega l^2}{2}$

Consider an element of length $dx$ as shown in the figure,


Emf developed across this element,

$dE=Bvl=B\left(\omega x\right)dx[\because v=\omega x]$

$\therefore$ Emf developed across the whole rod is,

$E=\int dE={\int }_{2l}^{3l}B\omega xdx$

$E=\frac{B\omega }{2}[9l^2-4l^2]=\frac{5B\omega l^2}{2}$

Hence, $\left(\text{C}\right)$ is the correct answer.